On Some Approximation Theorems for Power q-Bounded Operators on Locally Convex Vector Spaces

This paper deals with the study of some operator inequalities involving the power q-bounded operators along with the most known properties and results, in the more general framework of locally convex vector spaces.


Introduction
Let be a Hausdorff locally convex vector space over the complex field C. By calibration for the locally convex space we understand a family P of seminorms generating the topology P of , in the sense that this topology is the coarsest with respect to the fact that all the seminorms in P are continuous. Such a family of seminorms was used by the author and Wu [1] and many others in different contexts (see [2][3][4][5]).
It is well known that calibration P is characterized by the property that the set ( , ) = { ∈ : ( ) < } , > 0, ∈ P is a neighborhood subbase at 0. Denote by ( , P) the locally convex space endowed with calibration P.
Recall that a locally convex algebra is an algebra with a locally convex topology in which the multiplication is separately continuous. Such an algebra is said to be locally -convex (l.m.c.) if it has a neighborhood base U at 0 such that each ∈ U is convex and balanced (i.e., ⊆ for | | ≤ 1) and satisfies the property 2 ⊆ .
Any algebra with identity will be called unital. It is well known that unital locally -convex algebra A is characterized by the existence of calibration P such that each ∈ P is submultiplicative (i.e., ( ) ≤ ( ) ( ), for all , ∈ A) and satisfies ( ) = 1, where is the unit element.
An element of locally convex algebra A is said to be bounded in A if there exists ∈ C such that the set {( ) } ≥1 is bounded in A (see [6]). The set of all bounded elements in A will be denoted by A 0 .
Let C ∞ := C ∪ {∞} be the Alexandroff one-point compactification of C. Following Waelbroeck [7,8], we introduce the following. Definition 1. We call resolvent set in the Waelbroeck sense of an element from a locally convex unital algebra ( , P) the set of all elements 0 ∈ C ∞ for which there exists ∈ V 0 such that the following conditions hold: (a) the element − is invertible in , for any ∈ \ {∞}; (b) the set {( − ) −1 : ∈ \{∞}} is bounded in ( , P).
The resolvent set in Waelbroeck sense of an element will be denoted by ( ). The Waelbroeck spectrum of will be defined as 2 The Scientific World Journal
Definition 2. We say that a linear operator : → is -bounded (quotient-bounded) with respect to P if for any ∈ P there exists > 0 such that Denote by P ( ) the set which consists of all -bounded operators with respect to calibration P.
The set of all bounded elements in P ( ) will be denoted by ( P ( )) 0 (see [12]). It easily follows from [6, Proposition 2.14(ii)] that For ∈ ( P ( )) 0 we denote by ( ) the Waelbroeck resolvent set of and by ( ) the Waelbroeck spectrum of . The function is called the resolvent function of . It is well known that In this paper we evaluate the behaviour of the power of a -bounded operator from the algebra ( P ( )) 0 by some type of approximations. The main results have been announced in [14].

The Main Results
We continue to employ the notations from the previous sections and we will introduce two types of operatorial approximations for operators from the algebra ( P ( )) 0 which approximate a given operator on a convergent power bounded series. The power boundedness problem for operators acting on Banach spaces was largely developed in various frameworks by many authors (see [15][16][17]).
In the following, using the functional calculus from the ( P ( )) 0 algebra (see [7,8]), some important boundedness properties are obtained. Denote N * = N \ {0}. First we have the following.
for ∈ N * and for all ∈ C with | | > 1.
Proof. Assume that sup̂∈P̂( ) ≤ for ∈ N * . Since for | | > 1, then, by using the generalized binomial formula, we get from where we deducê for any ∈ N * and anŷ∈P. Therefore, the conclusion is verified.
Conversely, we have the following.
The last inequality was obtained by using Stirling's approximation. Now, for ∈ ( P ( )) 0 we introduce (see [18]) the following.
Next theorem shows how an operator from the ( P ( )) 0 algebra is related to its Yosida approximation. Proof. By evaluating ( , ) in terms of the resolvent ( , ), for | | > P ( ) we obtain from where it follows that the assertion of the theorem is true. Moreover, so (1) is true.
Proof. Property (i) implies P ( ) ≤ 1 so that the argumentation given in the proof of Theorem 7 implies that any ∈ C with | | > 1 belongs to the resolvent set of . Hence, using the generalized binomial formula, we get Now, by applying (i) again we obtain for anŷ∈P, whence by passing to supremum, the inequality (ii) holds. Conversely, (i) is a direct consequence of (ii).
Proof. From Theorem 8, for ∈ ( P ( )) 0 , is equivalent to The conclusion follows taking into account that for ∈ N * .

Application
Following [19], we see that the resolvent of is given by the Yosida approximation of is The above implies that is a contraction for ≤ 1.
It is clear that for estimating the powers of it seems to be better to use the Yosida approximation or Möbius approximation than the resolvent approximation.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.